Censored Stable Subordinators and Fractional Derivatives

Based on the popular Caputo fractional derivative of order $\beta$ in $(0,1)$, we define the censored fractional derivative on the positive half-line $\mathbb R_+$. This derivative proves to be the Feller generator of the censored (or resurrected) decreasing $\beta$-stable process in $\mathbb R_+$. We provide a series representation for the inverse of this censored fractional derivative, which we use to study general censored initial value problems. We are then able to prove that this censored process hits the boundary in a finite time $\tau_\infty$, whose expectation is proportional to that of the first passage time of the $\beta$-stable subordinator. We also show that the censored relaxation equation is solved by the Laplace transform of $\tau_\infty$. This relaxation solution proves to be a completely monotone series, with algebraic decay one order faster than its Caputo counterpart, leading, surprisingly, to a new regime of fractional relaxation models. Lastly, we discuss how this work identifies a new sub-diffusion model.